# Group delay

When thinking about filters, the filter amplitude frequency response is usually in the foreground of most considerations. After all – what we want from a filter is ‘selectivity’. But, like all other 2-port networks, filters also have a phase frequency response.  The filter transfer function value at a given frequency is a complex number – it has a magnitude and a phase angle. The resonators of a filter – if it is a bandpass filter – store the input energy for significant amounts of time, rather than allowing it to pass right through. Storage implies delay and the delay is frequency dependent.

The phase response of a bandpass filter is an important parameter for the transmission of signals through the filter. Unless the signal is just a sine-wave, the filter phase response will always create a finite amount of signal distortion. The time domain waveform of a signal at the input port of the filter will undergo a finite amount of change while it passes through the filter. The phase relationships of the signal frequency components are affected by the filter’s phase response.

A filter phase response is not linear because of a relationship between amplitude response and phase response. Given, that the amplitude response of a bandpass filter below and above the passband is usually very steep, this steepness is also felt in the phase response – especially towards the edges of the filter’s passband.

In other words: the filter phase is not linear vs frequency. But what is group delay now? It is a poorly chosen word for something relatively simple. Historically, it describes delay properties of a group of signals travelling through a transmission channel. The group delay response of a filter simply tells us how much a filter deviates from a linear phase response. A linear phase response implies flat (constant) group delay.  This is calculated by the derivative of the filter phase function with respect to the angular frequency (omega). The group delay value at a single frequency in the filter’s passband is just the time a sine-wave signal would take to pass through the filter and appear at the filter output port. The difference in group delay for signals spread out over a filter’s passband is a measure for the phase distortion introduced by the filter. Unlike in the time of analogue FDM (frequency division multiplex) many modern transmission modes or ‘modulations’ like OFDM are relatively immune to phase distortion, but filter phase distortion is still important as it affects the transmission quality (BER, MER) and needs to be kept to an acceptable minimum.

Filter group delay also has a relationship to the insertion loss of a bandpass filter. This can be understood by considering that the longer a signal is ‘kept’ in the filter, the more loss will occur. More signal amplitude swings: more current swings in the resonators: more losses.